Welcome to the world of Probability in Data Science! Let me start things off with an intuitive example. Suppose you are a teacher at a university. After checking assignments for a week, you graded all the students. You gave these graded papers to a data entry guy in the university and tell him to create a spreadsheet containing the grades of all the students.

Simulating data is a very powerful way to test your understanding of statistical concepts. We are going to use simulations to learn the basics of probability. First, we need to understand some different ways data might be distributed and how to simulate data from these distributions. A univariate distribution is the distribution of a single variable. The uniform distribution is the simplest distribution. All numbers in the range have an equal probability of being sampled. Use sample to sample from a discrete distribution.

Monte Carlo methods are used in corporate finance and mathematical finance to value and analyze complex instruments , portfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining the distribution of their value over the range of resultant outcomes. The advantage of Monte Carlo methods over other techniques increases as the dimensions sources of uncertainty of the problem increase. Monte Carlo methods were first introduced to finance in by David B. Hertz through his Harvard Business Review article, [3] discussing their application in Corporate Finance. In , Phelim Boyle pioneered the use of simulation in derivative valuation in his seminal Journal of Financial Economics paper.

For experienced modelers, the most challenging task in creating a simulation model is usually not identifying the key inputs and outputs, but selecting an appropriate probability distribution and parameters to model the uncertainty of each input variable. For example, Risk Solver software provides over 40 probability distributions -- so which one should you use? The answer depends on your application, but some general guidelines can be given.

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